Valeant Pharmaceuticals International Inc. and William Ackman are seeking a judge's help to speed up their $53 billion hostile takeover bid for Allergan Inc. in hopes of overturning the Botox maker's board before the company can strike a deal of its own.

Valeant and Mr. Ackman filed a lawsuit Friday—the latest legal front to be opened in the increasingly bitter fight—requesting that a judge order Allergan to hold a special meeting...